Existence of positive solutions for a system of Caputo fractional difference equations depending on parameters

R E S E A R C H

Open Access

Existence of positive solutions for a system

of Caputo fractional difference equations

depending on parameters

Shugui Kang

_{, Huiqin Chen and Jianmin Guo}

*_{Correspondence:} [email protected] School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, P.R. China

Abstract

We consider the existence of at least two positive solutions for a system of Caputo fractional difference equations

Cyj(t) = –

λ

ν

ν

to boundary conditionsyj(

ν

ν

ν

ν

j= 1,. . .,n. We use the Krasnosel’ski˘ı fixed point theorem to obtain the sufficient conditions of the existence of two positive solutions for this boundary value problem of Caputo fractional difference equations depending on parameters.

MSC: 26A33; 39A05; 39A12

Keywords: Caputo fractional difference; boundary value problem; fixed point theory; positive solution

1 Introduction

In this paper we consider a system of Caputo fractional difference boundary value problem (FBVP) of the form:

ν_Cjyj(t) = –λjfj

y(t+ν– ), . . . ,yn(t+νn– )

, (.)

yj(νj– ) =yj(νj+b) =yj(νj– ) = , (.)

wheret∈[,b+]N:={, , . . . ,b+},b> ,λj> ,  <νj≤,fj: [, +∞)×· · ·×[, +∞)→ [, +∞) are continuous functions for eachj(j= , , . . . ,n).ν_Cy(t) is the standard Caputo difference.

Fractional difference equations have been of great interest recently. It is caused by in-tensive development of the theory of discrete fractional calculus itself, see [–] and the references therein. Abdeljawad [] defined left and right Caputo fractional sums and dif-ferences, studied some of their properties. Holm [] introduced the fractional sum and difference operators. He developed and presented a complete and precise theory for com-posing fractional sums and differences. Atici and Sengül [] provided some analysis of discrete fractional variational problems, their paper also provided some initial attempts at using the discrete fractional calculus to model biological processes. Abdeljawad and Baleanu [] defined the right fractional sum and difference operators and obtained many of their properties. Then by using those properties they obtained a by-part formula anal-ogous to that in the usual fractional calculus. In [] the authors studied the stability of

discrete nonautonomous systems within the frame of the Caputo fractional difference by using the Lyapunov direct method. They discussed the conditions for uniform stability, uniform asymptotic stability, and uniform global stability. Mohammadi and Rezapour [] discussed the existence and uniqueness of solutions for some nonlinear fractional differen-tial equations via some boundary value problems by using fixed point results on ordered complete gauge spaces. Recently, Wu and Baleanu introduced some applications of the Caputo fractional difference to discrete chaotic maps in [, ].

In particular, the authors [–] developed some of the basic theory of fractional dif-ference both IVPs and BVPs with delta derivative on the time scale Z. In [], we ob-tained some results on the existence of one or more positive solutions for the Caputo fractional boundary value problems by means of cone theoretic fixed point theorems. Thus, the fractional difference equation has recently attracted increasing attention from a growing number of researchers. However, systems of discrete fractional boundary value problems are limited (see [–]). Among them, Atici and Eloe [] studied a linear system of fractional nabla difference equation with constant coefficients of the form

∇ν

y(t) =Ay(t) +f(t), t= , , . . . ,

where  <ν< ,Ais ann×nmatrix with constant entries, andf aren-vector valued func-tions. The operator∇ν

is a Riemann-Liouville fractional difference. They constructed the

fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system.

In [], the authors investigated the existence of solutions for ak-dimensional system of fractional finite difference equations:

ν_y

(t) +f

y(t+ν– ),y(t+ν– ), . . . ,yk(t+νk– )

= ,

ν_y

(t) +f

y(t+ν– ),y(t+ν– ), . . . ,yk(t+νk– )

= ,

. . . ,

νk_y

k(t) +fk

y(t+ν– ),y(t+ν– ), . . . ,yk(t+νk– )

= ,

y(ν– ) =y(ν+b) = ,

y(ν– ) =y(ν+b) = ,

. . . ,

yk(νk– ) =yk(νk+b) = ,

whereb∈N,  <νi≤,fi:Rk→Rare continuous functions fork= , , . . . . They

investi-gated the existence of solutions for thisk-dimensional system of fractional finite difference equations by using the Krasnosel’ski˘ı fixed point theorem.

In [], Goodrich studied the following pair of discrete fractional boundary value prob-lems:

–ν_y

(t) =λa(t+ν– )f

y(t+ν– ),y(t+νn– )

,

–ν_y

(t) =λa(t+ν– )f

y(t+ν– ),y(t+νn– )

y(ν– ) =ψ(y), y(ν– ) =ψ(y),

y(ν+b) =φ(y), y(ν+b) =φ(y),

wheret∈[,b]N:={, , . . . ,b},λ,λ> ,ν,ν∈(, ]. Goodrich obtained the existence of at least one positive solution to this problem by means of the Krasnosel’ski˘ı theorem for cons.

In [], the authors considered the existence of at least one positive solution to the dis-crete fractional system:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–ν_y

(t) =λf(t+ν– ,y(t+ν– ),y(t+ν– )), t∈[,b+ ],

–ν_y

(t) =λf(t+ν– ,y(t+ν– ),y(t+ν– )), t∈[,b+ ],

y(ν– ) =y(ν+b+ ) = ,

y(ν– ) =y(ν+b+ ) = ,

whereν,ν∈(, ].

Following this trend, in [], we discussed the boundary value problems of fractional difference system of the form

–νj_y

j(t) =λjfj

y(t+ν– ), . . . ,yn(t+νn– )

,

yj(νj– ) =ψj(yj), yj(νj+b) =φj(yj),

wheret∈[,b]N:={, , . . . ,b},λj> ,  <νj≤,fj: [, +∞)× · · · ×[, +∞)→[, +∞) are continuous functions. For eachjwe have thatψj,φj:Rb+→R(j= , , . . . ,n) are given

functions. We obtained the sufficient conditions for the existence of two positive solutions to the boundary value problem of a fractional difference system. In this paper we open our studies in this field. We establish some conditions on parametersλiwhich are able to

guarantee that FBVP (.)-(.) has at least two positive solutions and one positive solution, respectively, based on the Krasnosel’ski˘ı theorem.

This paper is organized as follows. In Section , we provide basic definitions and demon-strate some lemmas in order to prove our main results. In Section , we establish some results for the existence of at least two positive solutions to FBVP (.)-(.), and we con-clude with an example explicating our main result.

2 Preliminaries

In this section, we present some basic definitions in the discrete fractional calculus and establish some lemmas.

Definition .[] We define

t(ν)= (t+ )

(t+  –ν)

for anytandνfor which the right-hand side is defined. We also appeal to the convention that ift+  –νis a pole of the gamma function andt+  is not a pole, thent(ν)_{= .}

Definition .[] Theνth fractional sum of a functionf is defined by

–νf(t) = 

(ν)

t–ν

s=a

forν>  andt∈ {a+ν,a+ν+ , . . .}=Na+ν. We also define theνth Caputo fractional

difference forν>  by

ν_Cf(t) =–(n–ν)nf(t) = 

(n–ν)

t–(n–ν)

s=a

(t–s– )n–ν–n_f(s),

wheren–  <ν≤n.

Lemma .[, ] Assume thatν> and f is defined on domainsNa,then

–_a+(n–ν _ν)Cνf(t) =f(t) –

n–

k=

ck(t–a)k,

where ci∈R,i= , , . . . ,n– ;n–  <ν≤n.

Lemma .[] Let f :Na+ν×Na→Rbe given.Then

t–ν

s=a

f(t,s)

=

t–ν

s=a

tf(t,s) +f(t+ ,t+  –ν) for t∈Na+ν.

In order to get our main results, we now state an important lemma. This lemma gives a representation for the solution of (.)-(.), provided that the solution exists.

Lemma .[] Let <ν≤and g: [ν– ,ν– , . . . ,ν+b]Nν–→Rbe given.Then the solution of the FBVP

ν_Cy(t) = –g(t+ν– ), (.)

y(ν– ) =y(ν+b) =y(ν– ) =  (.)

is given by

y(t) =

b+

s=

G(t,s)g(s+ν– ),

where the Green’s function G: [ν– ,ν– , . . . ,ν+b]Nν–×[,b+ ]N→Ris defined by

G(t,s) = 

(ν) ⎧ ⎪ ⎨ ⎪ ⎩

(ν– )(t–ν+ )(ν+b–s– )ν–

– (t–s– )ν–_{, ≤s<t–ν+ ≤b+ ,}

(ν– )(t–ν+ )(ν+b–s– )ν–_{, ≤t–ν+ ≤s≤b+ .}

Remark Notice thatG(ν– ,s) = ,G(t,b+ ) = .Gcould be extended to [ν– ,ν+ b]Nν–×[,b+ ]N, so we only discuss on (t,s)∈[ν– ,ν+b]Nν–×[,b+ ]N.

Lemma .[] The Green’s function G satisfies the following conditions:

(i) G(t,s) > ,(t,s)∈[ν– ,ν+b]Nν–×[,b+ ]N.

(ii) maxt∈[ν–,ν+b]Nν–G(t,s) =G(ν+b,s),s∈[,b+ ]N.

(iii) minν+b

 ≤t≤(ν+b)G(t,s)≥



maxt∈[ν–,ν+b]Nν–G(t,s) =



The proofs of Lemma . and Lemma . can be found in [], so we omit their proofs. Let

Bj:=

yj: [νj– ,νj+b]Nν_j–→R,yj(νj– ) =yj(νj+b) =

_y

j(νj– ) = 

be equipped with the usual maximum norm · , it is easy to verify thatBjis the Banach

space. Then we putK:=B×B× · · · ×Bn. By equippingKwith the norm

(y, . . . ,yn)=y+· · ·+yn,

it follows that (K, · ) is a Banach space.

Now consider the operatorT:K→Kdefined by

T(y, . . . ,yn)(t, . . . ,tn) =

T(y, . . . ,yn)(t), . . . ,Tn(y, . . . ,yn)(tn)

, (.)

where we defineTj:K→Bjby

Tj(y, . . . ,yn)(tj) =λj b+

s=

Gj(tj,s)fj

y(s+ν– ), . . . ,yn(s+νn– )

. (.)

LetI:= [ν+b

 ,

(ν+b)

 ]× · · · ×[

νn+b

 ,

(νn+b)

 ]. In the sequel, we shall also make use of the

cone

:=

(y, . . . ,yn)∈K:y, . . . ,yn≥,

min

(t,...,tn)∈I

y(t) +· · ·+yn(tn)

≥

(y, . . . ,yn)

.

Lemma . Let T be the operator defined as in(.).Then T:→.

Proof We show first that for (y, . . . ,yn)∈K, by the definitionsTj(j= , , . . . ,n), it is clear

that

Tj(y, . . . ,yn)(tj)≥, j= , , . . . ,n.

On the other hand, we show that

min

(t,...,tn)∈I

T(y, . . . ,yn)(t) +· · ·+Tn(y, . . . ,yn)(tn)

≥ 

T(y, . . . ,yn)

for (y, . . . ,yn)∈K. In fact, by Lemma .(iii), we have

min

tj∈[ νj+b

 , (ν_j+b)

 ]

Tj(y, . . . ,yn)(tj)

≥ min

tj∈[ b+νj

 , (νj+b)

 ]

λj b+

s=

Gj(tj,s)fj

y(s+ν– ), . . . ,yn(s+νn– )

– (νj– )(νj+b–νj+ )(νj+b–s– )

νj–

+ (νj+b–s– )

νj–

×fj

y(s+ν– ), . . . ,yn(s+νn– )

= λj

(ν)

b+

s=

(νj– )(νj+b–s– )

νj–

–

(νj+b–s)(νj+b–s)

(b–s+ )(b–s+ ) –

(νj+b–s)

(b–s+ )

×fj

y(s+ν– ), . . . ,yn(s+νn– )

= λj

(ν)

b+

s=

(νj– )(νj+b–s– )

νj–

– (νj– )(νj+b–s) (b–s+ )(b–s+ )

×fj

y(s+ν– ), . . . ,yn(s+νn– )

= .

Finally, when ≤tj–νj+ ≤s≤b+ ,

Gj(tj,s) =



(νj)

(νj– )(tj–νj+ )(νj+b–s– )

νj– ,

then

_t

jGj(tj,s) = .

Therefore, we can get

Tj(y, . . . ,yn)(νj– ) = .

So the boundary conditions are satisfied, which completes the proof.

Finally, to accomplish proof of our main results, we state cones theory. In particular, we require the following well-known fixed point theorem for cons in [].

Theorem .[] LetBa Banach space and letK⊆Bbe a cone.Assume that and 

are bounded open sets contained inBsuch that∈ and ⊆ .Assume further that

T:K∩( \ )→Kis a completely continuous operator.If either

(i) Ty ≤ yfory∈K∩∂ andTy ≥ yfory∈K∩∂ ,or

(ii) Ty ≥ yfory∈K∩∂ andTy ≤ yfory∈K∩∂ ,

then the operator T has at least one fixed point inK∩( \ ). 3 Main results

In this section, we state and prove the existence of at least two positive solutions regard-ing FBVP (.)-(.). Then we conclude this section with examples to illustrate our main results. First, denote

αj= b+

s=

Gj(νj+b,s), βj=

[(νj+b)–νj+]

s=[νj+b–νj+] Gj

b–νj



+νj,s

ξ=

For convenience, we now present the conditions that we presume in the sequel.

=T(y, . . . ,yn)+· · ·+Tn(y, . . . ,yn)

FBVP(.)-(.)has at least two positive solutions,where

Proof Define the function

ϕj(r) =

r

nαjmax≤y+···+yn≤rfj(y, . . . ,yn)

, j= , . . . ,n.

It is easy to know thatϕj: (, +∞)→(, +∞) is a continuous function. From (H), we see

thatlim_r→fr

j(r)= , that is,limr→

r

nαjfj(r)= , and

 <ϕj(r) =

r

nαjmax≤y+···+yn≤rfj(y, . . . ,yn)

≤ r

nαjfj(r)

,

solimr→ϕj(r) = .

From (H), we see further that limr→∞ϕj(r) = . Then there exists r>  such that

ϕj(r) =maxr>ϕj(r) =λ∗j,j= , . . . ,n. For anyλj∈(,λ∗j), by the intermediate value

the-orem, there exist two pointsd∈(,r),d∈(r,∞) such thatϕj(d) =ϕj(d) =λj. Thus,

we have

fj(y, . . . ,yn)≤

d

nλjαj

, y+· · ·+yn∈[,d],

fj(y, . . . ,yn)≤

d

nλjαj

, y+· · ·+yn∈[,d].

On the other hand, since (H) and (H) hold, there existe∈(,d),e∈(d,∞) such

that

fj(y, . . . ,yn)

y+· · ·+yn≥

 nλjβj

, y+· · ·+yn∈(,e]∪

 e,∞

.

Thus

fj(y, . . . ,yn)≥

e

nλjβj

, y+· · ·+yn∈

 e,e

,

fj(y, . . . ,yn)≥

e

nλjβj

, y+· · ·+yn∈

 e,e

.

Application of Theorem . and Theorem . leads to two distinct positive solutions of FBVP (.)-(.) which satisfy

e≤(y, . . . ,yn)≤d, d≤y, . . . ,yn≤e.

The proof is complete.

By the proof of Theorem ., we obtain the following.

Corollary . If one of conditions(H)and(H)is satisfied,then for every <λj<λ∗j,

FBVP(.)-(.)has at least one positive solution.

Theorem . Suppose that(H), (H)hold.Then,for anyλj≥λ∗∗j ,FBVP(.)-(.)has at

least two positive solutions,where

λ∗∗_j =  nβj

inf

r>

r min

Proof Define the function

ψj(r) =

r

nβjminr/≤y+···+yn≤rfj(y, . . . ,yn)

, j= , . . . ,n.

We know thatψj: (, +∞)→(, +∞) is a continuous function. Forλj>λ∗∗j , there exists

 <e< +∞such that

fj(y, . . . ,yn)≥

e

nλjβj

, y+· · ·+yn∈

 e,e

.

By condition (H), there exists  <d<esuch that

fj(y, . . . ,yn)≤

d

nλjαj

, y+· · ·+yn∈[,d].

From condition (H), there existse<d< +∞such that

fj(y, . . . ,yn)

y+· · ·+yn≤

 nλjαj

, y+· · ·+yn∈[d, +∞).

LetMj=max≤y+···+yn≤dfj(y, . . . ,yn). Choosed>dsuch thatd≥λjMjαj. Then

fj(y, . . . ,yn)≤

d

nλjαj

, y+· · ·+yn∈[,d].

By Theorem . and Theorem ., the proof is complete.

From the proof of Theorem ., we get the following.

Corollary . Suppose that one of conditions(H)and(H)holds.Then,for everyλj>λ∗∗j ,

FBVP(.)-(.)has at least one positive solution.

Theorem . Suppose that one of the following cases is satisfied:

() (H), (H)hold,and <λj<_nαjLj;

() (H), (H)hold,and <λj<_nα

jlj.

Then FBVP(.)-(.)has at least one positive solution.

Proof () From (H) (namelylim(y+···+yn)→∞

fj(y,...,yn)

y+···+yn =Lj,tj∈[νj– ,νj+b]Nνj–, j= , , . . . ,n), for anyj> , there exists a numberR> , fory+· · ·+yn∈(R, +∞), we have

fj(y, . . . ,yn) < (Lj+j)(y+· · ·+yn).

LetMj=max≤y+···+yn≤Rfj(y, . . . ,yn). ChooseR>max{R,

Mj

Lj+j}, then fj(y, . . . ,yn) < (Lj+j)(y+· · ·+yn)

fory+· · ·+yn∈[,R]. As arbitrarily ofj, sofj(y, . . . ,yn)≤LjR. Note that  <λj<_nα

jLj, then

λ∗_j =  nαj

sup

R>

R

max≤y+···+yn≤Rf(y) > 

nαj

R LR=

 nαjLj

Namely  <λj<λ∗j. By means of Corollary ., FBVP (.)-(.) has at least one positive

solution.

The proof is similar to () and hence omitted.

Similarly, we have the following.

Theorem . Suppose that one of the following cases is satisfied:

() (H), (H)hold,and _nL

jβj <λj< +∞;

() (H), (H)hold,and _nl_jβj <λj< +∞.

Then FBVP(.)-(.)has at least one positive solution.

Theorem . Suppose that conditions(H)and(H)hold.Ifλjsatisfies_nβjLj <λj<

 nαjlj or

 nβjlj<λj<



nαjLj,then FBVP(.)-(.)has at least one positive solution.

Proof Suppose that  nβjLj <λj<



nαjlj holds. Choosej>  such that



nβj(Lj–j)≤λj≤

 nαj(lj+j). With condition (H), there existsτ>  such thatfj(y, . . . ,yn)≤(lj+j)(y+· · ·+yn) for

y+· · ·+yn∈(,τ). Thus, fory+· · ·+yn∈∂ τ, Tj(y, . . . ,yn)

= max

tj∈[νj–,νj+b]Nν_j– λj

b+

s=

Gj(tj,s)fj

y(s+ν– ), . . . ,yn(s+νn– )

≤λj(lj+j)(y+· · ·+yn) b+

s=

max

tj∈[νj–,νj+b]Nνj– Gj(tj,s)

=λj(lj+j)(y+· · ·+yn) b+

s=

Gj(νj+b,s)

≤λjαj

τ

nλjαj

=τ n

=

n(y, . . . ,yn),

j= , . . . ,n. That is,

T(y, . . . ,yn)(t, . . . ,tn)

=T(y, . . . ,yn)(t), . . . ,Tn(y, . . . ,yn)(tn)

=T(y, . . . ,yn)+· · ·+Tn(y, . . . ,yn)

≤ 

n(y, . . . ,yn)+· · ·+ 

n(y, . . . ,yn) =(y, . . . ,yn)

for (y, . . . ,yn)∈∂ τ.

By condition (H), there existsR>  such thatfj(y, . . . ,yn)≥(Lj–j)(y+· · ·+yn) for

LetR=max{τ,R}, for (y, . . . ,yn)∈∂ R, we get

By using Theorem ., we obtain the conclusion. A similar proof holds when 

nβjlj <λj<



nαjLj. The proof is complete.

We now present an example illustrating the sorts of boundary conditions that can be treated by Theorem ..

f,f: [, +∞)×[, +∞)→[, +∞), andyis defined on the time scale{–_,_, . . . ,_ },y

is defined on the time scale{–_,_, . . . ,_}.fandfsatisfy conditions of Theorem ..

A computation shows thatλ∗_≈.,λ∗_≈., then, for everyλj∈(,λ∗j) (j=

, ), problem (.) has at least two positive solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SK conceived of the study and participated in its design. HC drafted the manuscript and participated in its design and coordination. JG participated in the sequence correction. All authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. Project was supported by the National Natural Science Foundation of China (Grant No. 11271235) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111020).

Received: 4 November 2014 Accepted: 10 April 2015

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